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1.
European Journal of Business and Management
ISSN 2222-1905 (Paper) ISSN 2222-2839 (Online)
Vol.5, No.27, 2013
www.iiste.org
Measuring the Volatility in Ghana’s Gross Domestic
Product (GDP) Rate using the GARCH-type Models
Godfred Kwame Abledu(PhD)1* , Agbodah Kobina2
1.
School of Applied Science and Technology, Koforidua Polytechnic, PO Box 981, Koforidua, Ghana
2.
Statistics Department, Koforidua Polytechnic, PO Box 981, Koforidua, Ghana
* E-mail of the corresponding author: godfredabledu@gmail.com
Abstract
The objective of this paper was to empirically characterize the volatility in the growth rate of real Gross
Domestic Product (GDP) for Ghana in three sectors using data spanning from 2000 to 2012. The GARCH-type
models(GARCH, EGARCH and GJR-GARCH) were used for the analysis of data. The results of the study
present evidence that the symmetric GARCH(1, 1) structure applies reasonably well to GDP when quarterly
observations are used.. As expected from financial time series, the data for the study exhibit characteristics such
as leptokurtosis, clustering, asymmetric and leverage effects. It was found that there was a significant increase in
volatility and leverage effect.
1.
Introduction
The economy of Ghana has a diverse and rich resource base, and as such, has one of the highest GDP per capita
in Africa. Ghana is one of the top–ten fastest growing economies in the world, and the fastest growing economy
in Africa. Ghana remains somewhat dependent on international financial and technical assistance as well as the
activities of the extensive Ghanaian diaspora. Gold, timber, cocoa, diamond, bauxite, manganese, and many
other exports are major sources of foreign exchange. An oilfield which is reported to contain up to 3 billion
barrels (480×106 m3) of light oil was discovered in 2007. Oil exploration is ongoing and, the amount of oil
continues to increase.
Gross Domestic Product (GDP) is Ghana’s official measure of economic growth. There are three different
approaches that can be taken to calculate GDP; the production approach, the expenditure approach, and the
income approach. The approach used to calculate Ghana’s GDP on a quarterly basis is the production approach.
The Gross Domestic Product (GDP) in Ghana was worth 39.20 billion US dollars in 2011, according to a report
published by the World Bank. The GDP value of Ghana is roughly equivalent to 0.06 percent of the world
economy. Historically, from 1960 until 2011, Ghana’s GDP averaged 7.15 Billion USD reaching an all time high
of 39.20 Billion USD in December of 2011 and a record low of 1.20 Billion USD in December of 1960. The
gross domestic product (GDP) is a measure of national income and output for a given country's economy. The
gross domestic product (GDP) is equal to the total expenditures for all final goods and services produced within
the country in a stipulated period of time.
The reduction in the volatility of growth rates with country size is well known. Box and Jenkins(1976) uses size
measures to correct for possible heteroskedasticity in long run growth rates. Khan, et al.( 2011)calculate the
Spearman rank correlation coefficient of the volatility of GDP with total GDP across countries and argues that
the higher output variance of smaller countries is due to their greater openness and susceptibility to foreign
shocks. We argue that there is a highly structured relationship between aggregate output shocks and the size of
an economy and that microeconomic models should try to explain all of these empirical regularities.
2.
2.1.
Materials and Method
Procedure
The time series were first analysed to identify systematic patterns (frequency components or trends) which are
not salient in the time series. Autocorrelation and Crosscorrelation Functions (ACF, CCF) as well as Spectral
and Cross-Spectral densities were estimated for these purposes. All estimated correlation functions
(correlograms) were plotted with the 95% confidence intervals of consecutive lags in the specified range. The
sample autocorrelations had been used in the earlier part of the analysis to check the stationarity of the data set
and also to have a measure of the dependence considering the data as a time series.
65

2.
European Journal of Business and Management
ISSN 2222-1905 (Paper) ISSN 2222-2839 (Online)
Vol.5, No.27, 2013
www.iiste.org
Seasonal dependency was proved by comparing the results of the Partial Autocorrelation Function (PACF),
which considerably reduces the dependence on the intermediate elements, within the lag, and the results of the
ordinary ACF (Box and Jenkins, 1976; Box and Pierce, 1970). The software routines applied validate the
significance of the correlation coefficients rk by comparing their values to the standard error of rk, under the
assumption that the series is a white noise process and that all autocorrelations are equal to zero.
The development of ARIMA models is based on the methodology described in the classical work of Box and
Jenkins (1976). The procedure is applied separately to the landings and SST time series, as a univariate time
series approach, taking into account only the mathematical properties of the data, without involving the
biological or the physical background of the system. This kind of analysis supposes that other 'external factors'
do not participate in the process development or that their contribution is stochastic.
For each developed ARIMA model the standard three-steps procedure has been followed, namely model
identification, parameter estimation and finally the diagnosis of the simulation and its verification (Brockwell
and Davis, 1996). As mentioned above, the input series for ARIMA needs to be differenced to achieve
stationarity. The order of differencing is reflected in the d parameter. The general model introduced by Box and
Jenkins (1976) can be summarized by the use of the following three types of parameters: the autoregressive
parameters (p), the number of differencing (d), and moving average parameters (q).
In the notation introduced by Box and Jenkins, a model described as (0, 1, 2) means that it contains 0 (zero)
autoregressive (p) parameters and 2 moving average (q) parameters which were computed for the series after it
was differenced once. Similarly the required parameters sp, sd and sq of the seasonal ARIMA process are
determined according to the results of the corresponding ACF and PACF. The approach used consequently was
to estimate the seasonal model first, then study the residuals of this model to get a clearer view of the nonseasonal model involved. If the identification of the seasonal model was correct, these residuals showed the nonseasonal portion of the model.
After the identification of the tentative model, its parameters were estimated applying maximum-likelihood
methods. The final results include: the parameter estimates, standard errors, estimate of residual variance,
standard error of the estimate, log likelihood, Akaike's information criterion (AIC), Schwartz's Bayesian criterion
(SBC). The minimizing of SBC and AIC were used, taking into account both how well the model fitted the
observed series, and the number of parameters used in the fit.
2.2.
The GARCH Models
GARCH models are used as a successful treatment to the financial data which often demonstrate timepersistence, volatility clustering and deviation from the normal distribution. Among the earliest models is Engel
(1982) linear ARCH model, which captures the time varying
features of the conditional variance. Bollerslev (1986) develops Generalized ARCH (GARCH)
model, allowing for persistency of the conditional variance and more efficient testing. Engle and
Bollerslev (1986) invent the Integrated GARCH (IGARCH) model that provides consistent estimation under the
unit root condition. Engle et. al. (1987) design the ARCH-in- Mean (ARCH-M) model to allow for time varying
conditional mean. Nelson’s (1990b) Exponential GARCH (EGARCH) model allows asymmetric effects and
negative coefficients in the conditional variance function. The leveraged GARCH (LGARCH) model
documented in Glosten et. al. (1993) take into account the asymmetric effects of shocks from different
directions.
Since their introduction by Engle (1982), Autoregressive Conditional Heteroskedastic (ARCH) models and their
extension by Bollerslev (1986) to generalised ARCH (GARCH) processes, GARCH models have been used
widely by practitioners. At a ﬁrst glance, their structure may seem simple, but their mathematical treatment has
turned out to be quite complex. The aim of this article is to collect some probabilistic properties of GARCH
processes. Although the ARCH is simple, it often requires many parameters to adequately describe the volatility
process of an asset return some alternative models must be sought. Shrivastava, et al. (2010) and Hull(2006)
proposed a useful extension known as the generalized ARCH (GARCH) model. An important feature of
GARCH-type models is that the unconditional volatility σ depends on the entire sample, while the conditional
volatilities
Let
σt are determined by model parameters and recent return observations.
(ε t ) t ∈ℤ be
a sequence of independent and identically distributed (i.i.d.) random variables, and let
p ∈ ℕ = {1, 2, 3,..., ) and p ∈ ℕ o = ℕ ∪ {0} . Further, let α 0 > 0 , α1,..., α p −1 ≥ 0 , α p > 0 ,
66

3.
European Journal of Business and Management
ISSN 2222-1905 (Paper) ISSN 2222-2839 (Online)
Vol.5, No.27, 2013
β1,..., βq −1 ≥ 0 and β q > 0
process is
(σ t ) t ∈ℤ
www.iiste.org
be non-negative parameters. A GARCH(p, q) process
( X t ) t ∈ℤ
with volatility
is then a solution to the equations:
X t = σ tε t t ∈ℤ
(1)
,
p
q
σ t2 = α t + ∑α i X t2−1 + ∑ β jσ t2−1 t ∈ℤ
i =1
where the process
(σ t ) t ∈ℤ
(2)
,
j =1
is
non-negative. The sequence
(ε t ) t ∈ℤ
is referred to as the driving noise
sequence. GARCH (p, 0) processes are called ARCH (p) processes. The case of a GARCH (0, q) process is
excluded since in that case, the volatility equation (2) decouples from the observed process and the driving noise
sequence.
It is a desirable property that
σt
should depend only on the past innovations (ε t − h) h ∈ ℕ , that is, it is
measurable with respect to σ algebra generated by (ε t − h) h ∈ ℕ . If this condition holds, we shall call the
GARCH (p, q) process causal. Then
algebra σ (εt − h : h ∈ℕ0 ) ,
(εt + h)h∈ℕ0 , and Xt is independent
of σ (εt + h : h ∈ℕ ) , for fixed t. The requirement that all the coefficients α1,...,αp and β1 ,..., β q are non-
generated by
(εt −h)h∈ℕ0 .
( Xt ) is measurable with respect to σ
negative ensures that
Also,
σt
is
independent of
σ2 is non-negative, so that σt can indeed be defined as the square root of σ2 .
Equation(1) is the mean equation and is specified as an AR(p) process. Equation(2) is the
conditional variance equation and it is specified as the GARCH(1, 1) process. Conditional variance
models (Shrivastava, 2009), unlike the traditional or extreme value estimators, incorporate time varying
characteristics of second moment/volatility explicitly. By successively substituting for the lagged
conditional variance into equation(2), the following expression is obtained:
ht =
α0
∞
+ α1 ∑ i =1 β i −1ε t2−i
1− β
(3)
An ordinary sample variance would give each of the past squares an equal weight rather than
declining weights. Thus the GARCH variance is like a sample variance but it emphasizes the most
recent observations. Since ht is the one period ahead forecast variance based on past data, it is called
the conditional variance. The squared residual is given by:
vt = ε t2 − ht
(4)
Equation(4) is by definition unpredictable based on the past. Substituting equation(4) into
equation(2) yields an alternative expression as follows:
ε t2 = ω + (α1 + β )ε t2−1 + vt − β vt −1
(5)
It can immediately be seen that the squared errors followed an ARMA(1, 1) process. The
autoregressive root is the sum of α1 and β , and this is the rule which governs the persistence of
volatility shocks. The Autoregressive Moving Average (ARMA) Models have been used by many researcher
for forecasting(Shrivastava, et, al. , 2010; Abu and Behrooz, 2011) . Given a time series of data Z t , the ARMA
model is a tool for understanding and, perhaps, predicting future values in this series. The model consists of two
parts, an autoregressive (AR) part and “a” moving average (MA) part. The model is usually then referred to as
67

4.
European Journal of Business and Management
www.iiste.org
ISSN 2222-1905 (Paper) ISSN 2222-2839 (Online)
Vol.5, No.27, 2013
the ARMA (a, b) model where a is the order of the autoregressive part and b is the order of the moving average
part. The notation ARMA (a, b) refers to the model with “a” autoregressive terms and “b” moving-average
terms. This model contains the AR(a) and MA(b) models. A time series Z t follows an ARMA (1, 1) model if it
satisfies
a
b
Z1 = k + ωt + ∑ β i Z t −i + ∑ α iωt −i
i −1
(6)
i =1
where { ω t } is a white noise series. The above equation implies that the forecasted value is depended on the
past value and previous shocks. The notation MA(b) which refers to the moving average model of order b is
written as
b
Z t = k + ωt + ∑ α iωt −i
(7)
i =1
and the notation AR(a) which refers to the autoregressive model of order a, is as
a
Z1 = k + ωt + ∑ β i Z t −i
(8)
i −1
α1 ,...,α b are the parameters of the model, µ is the expectation of Zt (often assumed to equal to 0),
and the ωt ,...,ωt −b are again, white noise error terms.
where the
2.3.
Model Estimation and Evaluation
The forecast error is the difference between the realization and the forecast. Thus
eς =
x
^
(T+
ς )...-
x
T +ς
(9)
.
Assuming the model is correct, then we have
^
eς = E[ X T +ς ] + ε ς − xς
(10)
We investigate the probability distribution of the error by computing its mean and variance. One desirable
characteristics of the forecast
⌢
X T +ς is that it is unbiased. For an unbiased estimate, the expected value of the
forecast is the same as the expected value of the time series. Because
εt
is assumed to have a mean of zero, an
unbiased forecast implies E[ε ς ] . The fact that the noise is independent from one period to the next period
means that the variance of the error is:
^
Var[ε t ] = Var{E[ X T +ς ] − x T +ς } + Var[ε T +ς ] and σ ε 2 (ς ) = σ E 2 (ς ) + σ 2 .
(11)
The conditional-sum-of-squares is used to find starting values of parameters, then the maximum likelihood
estimate for the proposed models. The procedure for choosing these models relies on choosing the model with
the minimum AIC, AICc and BIC. The models are presented in Table 1 with their corresponding values of AIC,
AICc and BIC. Among those possible models, comparing their AIC, AICc and BIC as shown in Table 1,
ARIMA (1,1,1)(0,0,1)12 and ARIMA (1,1,2)(0,0,1)12 were chosen as the appropriate model that fit the data well.
The basic volatility measure follows recent work by Comin and Mulani (2005, 2006) and Comin and Philippon
(2005), among others. The measure of volatility is given by:
68

5.
European Journal of Business and Management
www.iiste.org
ISSN 2222-1905 (Paper) ISSN 2222-2839 (Online)
Vol.5, No.27, 2013
 n Z

i , j +τ 
 γ −γ
σit =  ∑ 

 i , j+τ

it


 t =i  Pit 
(
2.4.
1
2 2
)




(12)
Maximum Likelihood Method
As pointed out by Bera and Higgins (1993), the GARCH models are most often estimated
by maximum likelihood method. It is thus adopted in this study as well. The log likelihood function of the
GARCH model based on previous period’s information f t Ιψ t −1 ~ N (α 0 + α1 f t −1 , lt ) is given by
1 T
∑ li (θ )
T t −1
'
'
'
'
where θ = (ξ , γ ) with ξ and γ the conditional mean and conditional variance parameters respectively, and
l (θ ) =
1
ε2
lt (θ ) = const. − log( ht ) − t 2 The likelihood function provided above is maximized using Berndt, Hall,
2
2ht
Hall and Hausman (1974) numerical algorithm.
2.5.
Model Identification
This involves the determination of the order of the AR and MA for both seasonal and non-seasonal components.
This can be suggested by the sample ACF and PACF plots based on the Box-Jenkins approach. From Figures 1
and 2, the ACF plot tails of at lag 2 and the PACF plot spike at lag 1, suggesting that q= 2 and p=1 would be
needed to describe these data as coming from a non-seasonal moving average and autoregressive process
respectively.
Also looking at the seasonal lags, both ACF and PACF spikes at seasonal lag 12 and drop to zero for other
seasonal lags suggesting that Q= 1 and P= 1 would be needed to describe these data as coming from a seasonal
moving average and autoregressive process. Hence ARIMA (1,1,2)(1,0,1) could be a possible model for the
series.
3.
Results of the Study
3.1.
Data for the study
This study used data based on the quarterly real GDP in Ghana from three sectors(agriculture, industry and
services). The source is the Ghana Statistical Service’s Main Economic Indicators. The sample period is the first
quarter of 2000 through the fourth quarter of 2012. Each variable is seasonally adjusted. The quarterly growth is
calculated as (Yt − Yt −1 ) × 100 / Yt −1 , where Yt is the original data series (real GDP for agriculture, industry and
services) at time t. This statistical release contains independently compiled quarterly estimates of the gross
domestic product (GDP) for the period of first quarter of 1992 to second quarter of 2012. The estimates are
based on the 1993 System of National Accounts (SNA), International Standard Industrial Classification Revision
4 published by the United Nations and other international organizations and Quarterly National Accounts
Manual: Concepts, data sources, and compilation by International Monetary Fund (IMF). This means that the
methodology, concepts and classifications, are in accordance with the guidelines and recommendations of an
internationally agreed system of national accounts. The estimates of real GDP are expressed in terms of a 2006
base year.
Short-term indicators are used to estimate the quarterly GDP (ref Quarterly National Accounts Manual:
Concepts, Data sources, and Compilation - IMF) where Annual GDP estimates are calculated independently
from the quarterly estimates. Other than that, annual GDP estimates are derived as the sum of the GDP for the
four quarters. The quarterly value added and GDP estimates have been seasonally adjusted. Seasonal adjustment
is the process of estimating and removing seasonal effects from time series to reveal non-seasonal features. This
process is to provide a clearer view of short term movements and trends and also to allow earlier identification of
turning points
69

6.
European Journal of Business and Management
www.iiste.org
ISSN 2222-1905 (Paper) ISSN 2222-2839 (Online)
Vol.5, No.27, 2013
3.2.
Empirical Results
The descriptive statistics for the GDP series are shown in Table 2. Generally, there is a large difference between
the maximum and the minimum return of the index. The standard deviation is also high with regards to the
number of observations including a high level of fluctuation of the yearly GDPs. The mean is close to zero and
positive as is expected for a time series of returns.
There is also negative skewness, indicating an asymmetric tail which exceeds more towards negative values. The
GDP series are leptokurtic, given their large kurtosis statistics as shown in Table 2. The kurtosis exceeds the
normal value of three indicating that the return distribution is flat-tailed. Jarque and Bera(1980) test for
normality confirms the results based on skewness and kurtosis and the series are non-normal according to Jarque
and Bera which rejects normality test at 1% level of significance.
Figure 3 shows the GDP series from 2000 to 2012. Virtual inspection shows that volatility change over the time
and it tends to cluster with the periods of low volatility and periods of high volatility. The volatility is relatively
consistent from 2003 to the year 2009 and seems to increase in the middle of 2008 till 2010.
Table 3 shows the results of ARCH- LM test. This was done to see if there is any ARCH effect in the residuals.
The ARCH- LM test for the series shows a significant presence of ARCH effect with low p-value of 0.000. The
null hypothesis of no ARCH effect is rejected and a strong presence of ARCH effect is detected as is expected
for most financial time series. The test is conducted at different numbers of lags. Values in parenthesis indicate
the p-values. The zero p-value at all lags indicates the presence of ARCH effect in the series. Obs*R-squared is
the number of observations multiplied by the R-squared value. The results in Table 3 confirm that the GARCHtype models can be applied to the GDP series. In most empirical implementations, the values, p ≤ 2 and q ≤ 2,
are sufficient to model the volatility which provides a sufficient tradeoff between flexibility and parsimony
(Knight and Satchell, 1998). The symmetric GARCH and nonlinear asymmetric EGARCH, and GJR-GARCH
models were examined at different lags for p ≤ 2 and q ≤ 2.
Table 4 shows the results of GARCH model estimation. The AR order for the mean equation is selected by the
Akaike Information Criterion (AIC) criterion and is found to be three for all the sectors. The AIC is a measure of
the relative goodness of fit of a statistical model. In the general case, the AIC is equal to 2 k − 2 In ( L ) ,
where k is the number of parameters in the statistical model, and L is the maximized value of the likelihood
function for the estimated model.
The GARCH(1, 1), EGARCH(1, 1) and GJR-GARCH(1, 1) were found to be the most successful models
according to AIC. As they have the smallest value while satisfying restriction such as non-negativity for
symmetric GARCH. The models were estimated for the series using Quasi-Maximum likelihood assuming the
Gaussian normal distribution.
Data in Table 5 clearly indicates that the ARCH and GARCH terms are both significant for the service
sector with coefficient of 0.8811 and 0.0901 respectively. Similar parameter estimates are obtained for both
coefficients in the agricultural sector, whereas the estimates are quite discrepant in the industrial sector, with
relatively large estimates for ARCH (0.3682) and a small estimate for GARCH(0.4539). Moreover, the sum of
α1 and β1 , a parameter that shows the persistence of volatility is relatively high in the service and agricultural
sectors, but relatively low in the industrial sector. It should be noted that AR order for the mean equation
selected by the AIC criteria is found to be three for all data sets. The number in parentheses below the parameter
estimates are standard errors obtained from the heteroskedasticity consistent covariance matrix of the
parameters.
Table 6 gives the residual diagnostics corresponding to the estimates in Table 5. The Ljung-Box test is
used to check the autocorrelation of the residuals (Ljung and Box, 1979) and the Jarque-Bara test is used to
check the normality of the residuals(Jarque and Bara, 1987). The enteries in Table 5 are the p-values and
LB2(12) values. The LB2(12) is the Ljung-Box test of order 12 using squared standardized residuals. As Table 6
indicates, the null hypothesis of no autocorrelation is not rejected for all three sectors at 1% significance level.
70

7.
European Journal of Business and Management
www.iiste.org
ISSN 2222-1905 (Paper) ISSN 2222-2839 (Online)
Vol.5, No.27, 2013
The result of non-normality in residuals shows that the GARCH effect is insufficient to capture the
characteristics of the distribution.
The LB2 (12) is the Ljung-Box test of order 12 using squared standardized residuals and the normality test is
obtained from Jarque-Bera statistic. Entries represent corresponding p-values. P-values less than 0.05 imply the
hypothesis of remaining no ARCH effect is rejected and the hypothesis of normal distribution is rejected at the
5% level of significance.
The variance in the low volatility state is estimated at 0.4861 for the service sector, 0.4625 for the agricultural
sector, and 0.2534 for the industrial sector, and all these values are significant at the 5% significance level. The
variance in the high volatility state is estimated at 2.0811 for the service sector, 2.8925 for the agricultural sector,
and 1.2336 for the industrial sector, and all these values are significant at the 5% significance level. It is noted
that the variance in the high volatility state(St = 2) in the service, agricultural and industrial sectors are more than
four, six and eight times as great as the variances in the low volatility states respectively. It is then possible to
quantify the differences or the breaks in the variance of the GDP process reported in other studies(Kim, et.al.
2001; Bhar, et. al.2009)
4.
Conclusion
The ARCH and the GARCH models have been applied to a wide range of time series analysis but applications in
finance have been particularly successful and have been the focus of this paper. Recent studies have uncovered
evidence of a structural break in the variance of GDP process in many countries.
The paper used the GARCH – type models to characterize the volatility in the growth rate of real GDP in three
sectors. The main objects of interest were the unconditional volatility( σ )and conditional volatilities ( σt ). The
value that
σt
is set to can in some case make a large difference. For example, global volatility started picking
up with the advent of the 2007 crisis, peaking up in 2008. In such cases where there is a clear structural break in
volatility, the GARCH model experience difficulties since it is based on the assumption of average volatility
being constant.
Volatility models are estimated by maximum likelihood(ML)where parameter estimates are obtained by
numerically maximizing the likelihood function with an algorithm called the optimizer. Previous work has
documented the usefulness of a GARCH(1, 1) model without asymmetry in the innovation. In the absence of
market shocks GARCH variance will eventually settle to a steady state value. This is the value
2
t
σ 2 such that
2
σ = σ for all t.
References
Bera, A. and M. Higgins, (1993): ARCH models: properties, estimation and testing, Journal of Economic
Surveys, 7, 305-62.
Berndt, E., B. Hall, R. Hall and J. Hausman, 1974, Estimation and inference in nonlinear structural models,
Annals of Economic and Social Measurement, 4, 653-665.
Bollerslev, T., (1986): Combinationized autoregressive conditional heteroskedasticity. Journal of Econometrics,
31, 307-327.
Box, G. E. P. and Jenkins, G. M. (1976): Time series analysis: forecasting and control. Holden-Day, San
Francisco
Box, G. E. P. and Pierce, D. A. (1970): "Distribution of Residual Autocorrelations in Autoregressive-Integrated
Moving Average Time Series Models", Journal of the American Statistical Association, 65: 1509–1526.
Brockwell, P.J. and Davis, R.A. (1996): Time series: theory and methods(second edition) Springer-Verlag, New
York
Dhar, J. et. al. (2009): "Simulative approach to Constant Mean and Conditional Variance Heteroscedastic model
selection analysis using Likelihood Ratio Test for Indian Market Returns", Proc. IEEE Advance Computing
Conference 2009, Patiala.
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ISSN 2222-1905 (Paper) ISSN 2222-2839 (Online)
Vol.5, No.27, 2013
Figure 2: PACF of First Order Difference Series
Figure 3: GDP series from 1992 to 2012
75

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